Each of the four guests Heidegger invites over is a sort of riff on the same basic theme. They all squandered their good fortune and lost everything through their youthful foolishness. Given the chance for a classic do-over, they make the same mistakes again. Each of them does the same thing, the only difference being their particular vice of choice.
We'll start with the case of the Widow: she was once beautiful, but ruined her reputation – probably by enticing a lot of men a little too eagerly. After drinking the elixir, the Widow once again becomes vain, obsesses over her looks, and then encourages the three men to fight over her, just as she did when she was younger.
Mr. Medbourne, like the Widow, has fallen from his heyday on account of his foolishness. In this case, the vice we're looking at is greed. Mr. Medbourne was a successful businessman, but in his desire to make more and more money, he lost everything on risky investments (that's what the narrator means by "speculation"). When he gets the opportunity to experience youth a second time, Mr. Medbourne makes the same two mistakes he did the first time around: he obsesses over money, and he chases the Widow Wycherly. Tisk, tisk, Hawthorne would say.
In the case of Colonel Killigrew, we're dealing with a pleasure-seeker. Hawthorne, being Puritan and all, doesn't exactly outline on what "pleasures" Killigrew squandered his youth, but we're going to go out on a limb here and guess something along the lines of alcohol and women. It's hardly surprising, then, that this is what Killigrew returns to when he grows young again.
Last, but not least. Mr. Gascoigne is guilty of being power-hungry. He was at one point "a ruined politician, a man of evil fame," but is now so old that no one even remembers who he is anymore (1). Like all of Heidegger's other guests, Gascoigne is back to his bad ways the moment he feels that youthful spring in his step. The narrator even adds that he can't be sure if the politics on Mr. Gascoigne's mind belong "to the past, present, or future […] since the same ideas and phrases have been in vogue these fifty years" (28). Not only has Mr. Gascoigne failed to change his ways, but the world around him is similarly stagnated in vice.
We should say at this point that Heidegger's guests aren't exactly fleshed-out, fully developed characters. They function more as caricatures of different vices. These four clearly aren't as mysterious or compelling as Dr. Heidegger, which is certainly fine – there's still enough complexity to go around.
Speaking of complexity, the big question when it comes to the guests is whether or not they are genuinely transformed by the elixir. We argue both sides (and even find a third) in "Symbols, Imagery, Allegory," so check it out. As for the fate of these four foolish folk – namely the fact that they learned absolutely nothing from their experiences – check out "What's Up with the Ending?"